You presented an analysis of yours, but did not really describe its current state beyond "being broken" :-) What does it do currently?

This might be a case where the USB download mode might come handy. But before your hopes get high: That will probably take some time and careful evaluation. While geekmaster (plus probably others, e.g. yifanlu) has done lots of work on this matter, his attempts were directed at newer kindles. Nevertheless, the "Fastboot Manifesto" thread has first hints for the K3, too.

I've brought my K3 successfully into USB downloader mode, but then failed miserably trying to dump the (e)MMC data. That was just a short try to test things, and I stopped playing with it at that point.

At the current point, it will probably need much more investigation and maybe some development work on a uboot for K3 USB downloader mode. This might take a lot of time, maybe it won't ever happen (sorry).

Another variant might be to use serial console access (on the K3, you can build a customized connector if you don't want to pop the cover open). It also will need some experience, but that route is described in some threads in this forums. I don't remember exactly, how to exactly approach the problem then. But in the end it's writing a working image into the system partition of the eMMC (what you called "ROM" :-) ).